A no-go theorem usually starts from the conventional probability theory, which is widely regarded as the true representation of logical deduction, and assumes certain additional plausible physical premises: realism, locality, no superdeterminism, observer independence, etc. One then shows that such a model leads to prediction which is contradicted by quantum mechanics. Hence, one concludes that, at least one of the assumptions, or the rules of conventional probability must be violated by quantum mechanics. Let's denote the physical assumption that a no-go theorem tries to prove to be violated by quantum theory as \(\mathcal {W}\) . For instance, \(\mathcal {W}\)  could be “measured facts are observer independent". The no-go theorem may be constructed independent of the underline physical theory. But if the logical deduction in the proof of theorem utilizes the law of total probability in one of the forms of (REF ), (REF ), or (REF ) without calling out the appropriate sufficient condition \(C\) , then we know the resulting statement (could be in a form of inequality) will not hold in quantum theory. This leaves a loophole in the logical deduction. Because the contradiction shown in the no-go theorem could be just due to the fact that \(C\)  is not met in the experiment setup, instead of the intended conclusion that \(\mathcal {W}\)  is violated by quantum theory. Thus, the no-go theorem does not reach the conclusion as desired. We will examine several such no-go theorems in this sectionWe do not include the no-go theorem [1]} widely discussed in the literature since its proof does not invoke the law of total probability..
